As I’ve written elsewhere, I sometimes think that generations of scientists raised in space might help advance physics, having lived in a world dominated by inertia (rather than friction). Especially in regard to a visceral understanding of microgravity. Like characters in the TV series The Expanse.

So, this Space.com article (June 18, 2018) “Relativity: The Thought Experiments Behind Einstein’s Theory” grabbed my attention. Another historical exposition by Paul Sutter.^{1}

Look at these two equations:

**F** = *m***a**

Both include the mass of an object. In the first case as we experience everyday, it takes, for example, twice the force to accelerate a mass twice as much. In the second case, the masses appear to act like electric charges in Coulomb’s Law:

In one case, we get a notion of pushing in order to accelerate or decelerate an object (or action as a result of contact between objects [locality]). Doing work on something, applying energy to something in order to overcome inertia. Hence, an object’s **inertial mass**.

In the other, we get a notion of objects just naturally attracting or “falling” toward each other (or action without contact — action at a distance — between objects [nonlocality]) depending on their **gravitational mass**.

Sutter explores how Einstein thought about the implications of these two perspectives. His article includes two video segments on the topic.

Objects with twice the mass feel twice the attraction toward the Earth, and should therefore fall twice as quickly. But years back, Galileo Galilei had conclusively shown that they don’t: Neglecting air resistance, all objects fall at the same rate regardless of their mass.

Thus for Newton’s theory to work,

inertial mass had to be the same as gravitational mass, but only by sheer coincidence: there was noreasonfor this equality to hold. For an object with twice the mass, the Earth may pull on it twice as strongly, but this is perfectly canceled out by the fact that it’s now twice as hard to get the object moving.Inertial and gravitational masses move in perfect lockstep.To Einstein, this was a major clue. Lurking in the shadows of gravity was his precious

special relativityand the essential concept ofspace-time, and what made that realization possible was the elevation of theequivalencebetween inertial and gravitational masses into a fundamentalprinciple, rather than the awkward afterthought it had been.The implication is clear (or at least, it was clear to Einstein):

Gravity causes acceleration, and acceleration causes gravity. They are absolutely identical.

Our everyday notion of Euclidian space had to fall in favor of a curved geometry.

The geometry that describes this relationship simply isn’t the normal Euclid-derived stuff that we were taught in high school. It’s non-Euclidean, or the geometry of curved spaces.

Einstein not only re-visualized our notion of space and time but developed the mathematics into a comprehensive theory.

**UPDATE** 6/29/2018: Sutter continued his exposition in another Space.com article (June 28, 2018), “Why Relativity’s True: The Evidence for Einstein’s Theory.” His article includes two more video segments on the topic.^{2}

His focus this time is on evidence supporting Einstein’s theory, as well as the thought experiments which anticipated corroboration.

- Explanation of Mercury’s orbit
- Detection of light bending around the sun during total solar eclipses
- Redshift of light traveling upward from the surface of the Earth

But in all regards, GR passes with flying colors; from sensitive satellites to gravitational lensing, from the orbits of stars around giant black holes to ripples of gravitational waves and the evolution of the universe itself, Einstein’s legacy is likely to persist for quite some time.

[1] *Paul Sutter** is an astrophysicist at **The Ohio State University** and the chief scientist at **COSI science center**. Sutter is also host of “**Ask a Spaceman**” and “**Space Radio**,” and leads **AstroTours** around the world. Sutter contributed this article to **Space.com’s Expert Voices: Op-Ed & Insights**.*

[2] Sutter’s merry-go-round example of relativistic shrinking and therefore introduction of non-Euclidian (curved) geometry left me uneasy (as well as the lack of visualization). He described a rotating circular ring of individual model horses nose-to-tail so that there were no gaps between each one. Then he described how in passing horizontally past our view at near the speed of light, the length of each horse is shortened so that gaps arise between them. Hence, the circumference is altered, and the relationship between the ring’s diameter and circumference (pi) no longer is that of Euclidian space.

Did he mix two frames of reference?

Consider a solid rotating ring or a continuous rotating band in the same situation. Even such a band holding all Sutter’s horses together nose-to-tail. No gaps will appear in that band as it rotates near the speed of light, eh. Or even such a band with pictures of horses painted in the same fashion on the surface. Gaps will not appear between those figures.

As noted on Wiki (below), “object” is merely a distance between endpoints mutually at rest. **For someone on the merry-go-round**, everything’s at rest. Spacing of the model horses will remain the same (even if there are actually micro gaps between their noses and tails). And standard rulers in the rotating frame will appear unchanged as well.

However, if (prior to rotation) standard rulers were placed (end-to-end) around the circumference (periphery) of the merry-go-round — but not on the merry-go-round — so as to be visible to an observer on the merry-go-round, then that observer will notice the rulers become contracted (shorter). But would there be gaps between them?

**For an observer outside the rotating frame**, things will appear different. But from what perspective? From far enough away, will the size of the ring change? Not its diameter because its radius always is perpendicular to its motion (as noted below). However, its circumference will appear Lorentz-contracted to a smaller value than at rest (in the rest frame of someone on the merry-go-round).

Considering the circumference as small line segments (as noted below), then close up, so that only a quite small arc of the ring is visible as it rotates by us — a single horse crosses in front of us, the length of that arc will shrink. But will there be gaps between the model horses? Again, as already noted, distances between any endpoints on (or microscopically near) those horses’ noses or tails will shrink as well.

So, while Sutter’s conclusion likely stands, his explanation is confusing.

References

[Wiki] **Lorentz length contraction**

Length contractionis the phenomenon that a moving object’s length is measured to be shorter than its proper length, which is the length as measured in the object’s own rest frame.First it is necessary to carefully consider the methods for measuring the lengths of resting and moving objects.

^{}Here, “object” simply means a distance with endpoints that are always mutually at rest,i.e., that are at rest in the same inertial frame of reference.It turns out that the proper length remains unchanged and always denotes the greatest length of an object, and the length of the same object measured in another inertial reference frame is shorter than the proper length.

This contraction only occurs along the line of motion…The principle of relativity (according to which the laws of nature must assume the same form in all inertial reference frames) requires that

length contraction is symmetrical: If a rod rests in inertial frame S, it has its proper length in S and its length is contracted in S’. However, if a rod rests in S’, it has its proper length in S’ and its length is contracted in S.

Stack Exchange: **What happens if a super fast rotating ball accelerates near speed of light?**

The geometry for a spinning ball changes. If you consider the circumference as small line segments, the fact that they are spinning means that the line segments are moving in the direction of motion and exhibit Lorentz contraction while

the radii are not foreshortened since they are moving perpendicular to the spin. You are now dealing with non Euclidean geometry for the sphere sincethe circumference is no longer equal to pi times the diameter. [November 14, 2015]

University of Illinois Department of Physics: **Q & A: relativistic merry-go-round**

This situation has practical consequences. Storing a large number of bunches of charged particles in a

circular storage ringand then accelerating them to high energies involves this effect. Typically, the rings of magnets in a modern synchrotron are fixed in radius and the radio-frequency cavities are fixed in frequency and their spacing.The charged particles travel at nearly the speed of light all the time, so their travel times do not change much as the energy is raised from immense to really immense. Nor does the spacing of the bunches around the ring. What changes though, is that in one moving bunch’s frame, the neighboring bunches get farther apart as the energy is increased.This has an effect on the electrostatic force one bunch exerts on another as the energy increases (they go down. Real accelerators have more troubles with residual electromagnetic fields oscillating in the metal beampipe).In the frame of one of the bunches, the distance to the next has increased, but the same number of bunches stay equally spaced around the ring, so the whirling observer thinks the circumference has increased. But, paradoxically, it takes less of his time to go around that circle at approximately the same speed. (This is observed when putting particles with known lifetimes, such as muons, into these storage rings — they make more turns around the ring on average before decaying.)[10/22/2007]

[Wiki] **Ehrenfest paradox**

The Ehrenfest paradox concerns the rotation of a “rigid” disc in the theory of relativity.

In its original formulation as presented by Paul Ehrenfest 1909 in relation to the concept of Born rigidity within special relativity, it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry.

The radius R as seen in the laboratory frame is always perpendicular to its motion and should therefore be equal to its value R0 when stationary. However, the circumference (2πR) should appear Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the contradiction that R=R0 and R<R0.

The paradox has been deepened further by Albert Einstein, who showed that since measuring rods aligned along the periphery and moving with it should appear contracted, more would fit around the[at rest]circumference, which would thus measure greater than 2πR. This indicates that geometry is non-Euclidean for rotating observers, and was important for Einstein’s development of general relativity.

Paul Halpern (physicist and science writer and author of

The Quantum Labyrinth: How Richard Feynman and John Wheeler Revolutionized Time and Reality) discusses this topic in a May 17, 2015, article “Life in a Freely Falling Elevator.”Wiki: Free fall

Another example, “These 5 mind-melting thought experiments helped Albert Einstein come up with his most revolutionary scientific ideas” from July 27, 2016.

As to the mathematics of gravity in General Relativity, Wiki:

Wiki:

Sean Carroll’s Mindscape Podcast with Carlo Rovelli “Episode 2: Carlo Rovelli on Quantum Mechanics, Spacetime, and Reality” (July 10, 2018) is an excellent dialog on current theories of gravity like string theory and loop quantum gravity.

In particular, I found the characterization / clarification of the notion of

spacetimefascinating.And as to equations and how gravity behaves at extremely short distances: “Since

we don’t have a reliable theory of how gravity behaves at short distancesI’ve just sketched in the gravity line roughly.” — The Lightness of Being: Mass, Ether, and the Unification of Forces by Frank WilczekThis MinutePhysics YouTube video also is interesting: “Our Ignorance About Gravity” (published on Jun 20, 2019).

And from the video’s transcript:

And as to the speed of so-called gravitational waves, Ethan Siegel (veteran science communicator) discusses the reasons that such waves travel at the speed of light, the same speed as other forms of electromagnetic radiation in this Forbes article, “Ask Ethan: Why Do Gravitational Waves Travel Exactly at the Speed of Light?” (Jul 6, 2019). He even includes a table of equations!

Siegel cites research on binary pulsar systems as another reason.

This Washington Post article “Everything you thought you knew about gravity is wrong” (August 2, 2019) by science communicator Richard Panek unpacks illusions about each word in the answer, “Gravity is the

forceofattractionthat makes thingsfallstraight down.”There is, as yet, no quantum solution to gravity.

Regarding the two 1/r^2 equations at the beginning of this post, Wiki’s article on point particle (mass, charge) is interesting: